In this paper, we consider the Lane–Emden problem [Formula: see text] where Ω is a bounded domain in ℝN and p > 2. First, we prove that, for p close to 2, the solution is unique once we fix the projection on the second eigenspace. From this uniqueness property, we deduce partial symmetries of least energy nodal solutions. We also analyze the asymptotic behavior of least energy nodal solutions as p goes to 2. Namely, any accumulation point of sequences of (renormalized) least energy nodal solutions is a second eigenfunction that minimizes a reduced functional on a reduced Nehari manifold. From this asymptotic behavior, we also deduce an example of symmetry breaking. We use numerics to illustrate our results.

We consider the Lane-Emden Dirichlet problem −∆u = |u| p−1 u, in B,where p > 1 and B denotes the unit ball in R 2 . We study the asymptotic behavior of the least energy nodal radial solution u p , as p → +∞. Assuming w.l.o.g. that u p (0) < 0, we prove that a suitable rescaling of the negative part u − p converges to the unique regular solution of the Liouville equation in R 2 , while a suitable rescaling of the positive part u + p converges to a (singular) solution of a singular Liouville equation in R 2 . We also get exact asymptotic values for the L ∞ -norms of u − p and u + p , as well as an asymptotic estimate of the energy. Finally, we have that the nodal line N p := {x ∈ B : |x| = r p } shrinks to a point and we compute the rate of convergence of r p .

ABSTRACT. Assuming B R is a ball in R N , we analyze the positive solutions of the problemthat branch out from the constant solution u = 1 as p grows from 2 to +∞. The non-zero constant positive solution is the unique positive solution for p close to 2. We show that there exist arbitrarily many positive solutions as p → ∞ (in particular, for supercritical exponents) or as R → ∞ for any fixed value of p > 2, answering partially a conjecture in [12]. We give the explicit lower bounds for p and R so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.

We study the nodal solutions of the Lane Emden Dirichlet problem −∆u = |u| p−1 u, in Ω,where Ω is a smooth bounded domain in R 2 and p > 1. We consider solutions u p satisfyingand we are interested in the shape and the asymptotic behavior as p → +∞.First we prove that (*) holds for least energy nodal solutions. Then we obtain some estimates and the asymptotic profile of this kind of solutions. Finally, in some cases, we prove that pu p can be characterized as the difference of two Green's functions and the nodal line intersects the boundary of Ω, for large p.2000 Mathematics Subject Classification. Primary: 35J91; Secondary: 35B32. Key words and phrases. superlinear elliptic boundary value problem, least energy nodal solution, asymptotic behavior, variational methods.This work has been done while the second author was visiting the Mathematics Department of the University of Roma " Sapienza" supported by INDAM-GNAMPA. He also acknowledges the national bank of Belgium and the program "Qualitative study of solutions of variational elliptic partial differerential equations. Symmetries, bifurcations, singularities, multiplicity and numerics" of the FNRS, project 2.4.550.10.F of the Fonds de la Recherche Fondamentale Collective for the partial support.

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